Perfect State Transfer on Abelian Cayley Graphs

TL;DR

This paper characterizes when perfect state transfer occurs in connected abelian Cayley graphs, unifying previous results for circulant and cubelike graphs and providing new insights into quantum information applications.

Contribution

It offers a unified characterization of PST in abelian Cayley graphs, generalizing prior results and solving open problems in the field.

Findings

Many previous results on circulant and cubelike graphs are derived or generalized.

New conditions for PST in abelian Cayley graphs are established.

Several open problems in the area are answered.

Abstract

Perfect state transfer (PST) has great significance due to its applications in quantum information processing and quantum computation. In this paper we present a characterization on connected simple Cayley graph $\Gamma={\rm Cay}(G,S)$ having PST. We show that many previous results on periodicity and existence of PST of circulant graphs (where the underlying group $G$ is cyclic) and cubelike graphs ($G=(\mathbb{F}_2^n,+)$) can be derived or generalized to arbitrary abelian case in unified and more simple ways from our characterization. We also get several new results including answers on some problems raised before.